Transmission apparatus and error correction method

ABSTRACT

A transmission apparatus includes, a receiving circuit that receives a reception signal indicating a coded bit string, a decoding circuit that decodes and corrects the bit string by using a spatially-coupled low density parity check code constituted by arranging element matrixes stepwise in a diagonal direction, a parity check matrix of the spatially-coupled low density parity check code including at least one element matrix having at least one of a number of rows and a number of columns different from a number of rows and a number of columns of other element matrixes when each sparse matrix constituting the parity check matrix is regarded as an element matrix, and outputs the corrected bit string.

CROSS-REFERENCE TO RELATED APPLICATION

This application is based upon and claims the benefit of priority of theprior Japanese Patent Application No. 2017-156665, filed on Aug. 14,2017, the entire contents of which are incorporated herein by reference.

FIELD

The embodiments discussed herein are related to a transmission apparatusand an error correction method.

BACKGROUND

In the field of transmission apparatuses, there have been known varioustechniques for improving the characteristics of error correction processfor correcting errors in a receive signal. For example, there has beenproposed a technique in which a cyclic permutation matrix is arranged soas to satisfy a predetermined weight distribution, and a low densityparity check matrix formed to gradually increase the weight of a row isused to achieve efficient erasure correction of data having a short codelength (see, e.g., International Publication Pamphlet No. WO2006/106841). In addition, there has been proposed a technique in whichthe size of a region of a root matrix is extended and then a nonzeroelement is moved so that the density of the nonzero element becomesuniform in the extended region to generate a desired parity check matrixat a high speed (see, e.g., Japanese Laid-open Patent Publication No.2015-103866). Further, there has been proposed a technique for improvingthe error correction characteristics by setting a large column weight inthe column direction of an element matrix corresponding to the end ofthe bit string of a signal among element matrixes of a spatially-coupledlow density parity check code (see, e.g., Japanese Laid-open PatentPublication No. 2016-213701).

It may be possible to improve the error correction characteristics byusing the parity check matrix having the large column weight in thecolumn direction of the element matrix corresponding to the end of thebit string of the signal. In this aspect, however, the parity checkmatrix will contain parts with different column weights. When adifference in column weights in the parity check matrix gets larger, ashift in processing time due to the column weight difference may occur.Thus, a circuit for executing an error correction process may not beeasily implemented by a semiconductor device or the like.

Related techniques are disclosed in, for example, InternationalPublication Pamphlet No. WO 2006/106841, Japanese Laid-open PatentPublication Nos. 2015-103866 and 2016-213701, and Sarah Johnson andGottfried Lechner, “Spatially Coupled Repeat-Accumulate Codes”, IEEECOMMUNICATIONS LETTERS, VOL. 17, NO. 2, February 2013.

SUMMARY

According to an aspect of the embodiments, a transmission apparatusincludes, a receiving circuit that receives a reception signalindicating a coded bit string, a decoding circuit that decodes andcorrects the bit string by using a spatially-coupled low density paritycheck code constituted by arranging element matrixes stepwise in adiagonal direction, a parity check matrix of the spatially-coupled lowdensity parity check code including at least one element matrix havingat least one of a number of rows and a number of columns different froma number of rows and a number of columns of other element matrixes wheneach sparse matrix constituting the parity check matrix is regarded asan element matrix, and outputs the corrected bit string.

The object and advantages of the invention will be realized and attainedby means of the elements and combinations particularly pointed out inthe claims. It is to be understood that both the foregoing generaldescription and the following detailed description are exemplary andexplanatory and are not restrictive of the invention, as claimed.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1A is a view illustrating an example of a spatially-coupled lowdensity parity check code;

FIG. 1B is a view illustrating an example of element matrixesillustrated in FIG. 1A;

FIG. 2 is a view illustrating an example of a spatially-coupled lowdensity parity check code according to an embodiment;

FIG. 3 is a view illustrating an example of a spatially-coupled lowdensity parity check code according to a first embodiment;

FIG. 4 is a view illustrating an example of a spatially-coupled lowdensity parity check code according to a second embodiment;

FIG. 5 is a view illustrating an example of a spatially-coupled lowdensity parity check code according to a third embodiment;

FIG. 6 is a view illustrating an example of a spatially-coupled lowdensity parity check code according to a fourth embodiment;

FIG. 7 is a view for explaining a method of generating thespatially-coupled low density parity check code illustrated in FIG. 6;

FIG. 8 is a view illustrating an example of a protograph;

FIG. 9A is a view illustrating a communication system;

FIG. 9B is an internal circuit diagram of a transmitter illustrated inFIG. 9A;

FIG. 9C is an internal circuit diagram of a receiver illustrated in FIG.9A;

FIG. 9D is a view illustrating a storage system;

FIG. 10 is a view illustrating a spatially-coupled RA code according toan embodiment;

FIG. 11 is a view for explaining a method of generating aspatially-coupled RA code 5 illustrated in FIG. 10; and

FIG. 12 is a view illustrating another example of the protograph.

DESCRIPTION OF EMBODIMENTS

Hereinafter, a transmission apparatus and an error correction methodaccording to embodiments will be described with reference to thedrawings. It is to be, however, noted that the technical scope of thepresent disclosure is not limited to these embodiments.

(Spatially-Coupled Low Density Parity Check Code)

FIG. 1A is a view illustrating an example of a spatially-coupled lowdensity parity check code and FIG. 1B is a view illustrating an exampleof element matrixes illustrated in FIG. 1A.

A spatially-coupled low density parity check (LDPC) code 100 is a matrixused to detect an error when a received signal is decoded and correctthe detected error. The spatially-coupled LDPC code 100 includes elementmatrixes 101 that are arranged W by W adjacent to each other in a rowfrom the upper left end to lower right end of the spatially-coupled LDPCcode 100. That is, the spatially-coupled low density parity check code100 is constituted by the element matrixes 101 arranged stepwise in thediagonal direction. In the spatially-coupled LDPC code 100, a so-calledzero matrix having a value of “0” is arranged in an area where noelement matrix 101 is arranged.

At the left end of the spatially-coupled LDPC code 100, W elementmatrixes 101 denoted by H_(1,1) to H_(1,W) are sequentially arranged inthe column direction with the element matrix 101 denoted by H_(1,1)located at the upper left end of the spatially-coupled LDPC code 100. Welement matrixes 101 denoted by H_(2,1) to H_(2,W) are sequentiallyarranged in the column direction on the right side of the W elementmatrixes 101 denoted by H_(1,1) to H_(1,W). The upper end element matrix101 denoted by H_(2,1) is located on the same row as the element matrix101 denoted by H_(1,2). W element matrixes 101 denoted by H_(3,1) toH_(3,W) are sequentially arranged in the column direction on the rightside of the W element matrixes 101 denoted by H_(2,1) to H_(2,W). Theupper end element matrix 101 denoted by H_(3,1) is located on the samerow as the element matrix 101 denoted by H_(2,2).

Each of the element matrixes 101 is arranged to be shifted downward froman element matrix 101 arranged in the column adjacent to the left sideby the number of rows of the adjacent element matrix 101. At the rightend of the spatially-coupled LDPC code 100, W element matrixes 101denoted by H_(L,1) to H_(L,W) are arranged with the upper end elementmatrix 101 denoted by H_(L,1) located on the same row as the elementmatrix 101 denoted by H_((L−1),2). The lower end element matrix 101denoted by H_(L,W) is located at the lower right end of thespatially-coupled LDPC code 100.

Each of the element matrixes 101 is a 3×6 matrix. In general, an elementmatrix is a matrix having a size of length×width=(N−K)×N in which “0”and “1” are arranged. In the example illustrated in FIG. 1B, N=6 andK=3, but in typical, each of K and N is a natural number of hundreds tothousands, and K<N. In the spatially-coupled LDPC code 100, all theelement matrixes 101 have the same size. That is, in each of the elementmatrixes 101, K and N are the same number.

The number of “1” in the column direction of the element matrix 101 iscalled column weight. In the element matrix 101, the column weight ofthe first column indicated by an arrow A is “2”, the column weight ofthe second column indicated by an arrow B is “3” and the column weightof the third column indicated by an arrow C is “2”. The column weight ofthe fourth column indicated by an arrow D is “1”, the column weight ofthe fifth column indicated by an arrow E is “1” and the column weight ofthe sixth column indicated by an arrow F is “2”.

When the spatially-coupled LDPC code 100 is used to execute an errorcorrection process, by increasing the column weight of the elementmatrix 101, the accuracy of error correction, that is, thecharacteristics may be improved. However, when the column weight of theelement matrix 101 is increased, the process load of the errorcorrection process increases, which may result in increase in scale of aprocessing circuit for executing the error correction process.

(Spatially-Coupled LDPC Code According to Embodiment)

FIG. 2 is a view showing an example of a spatially-coupled LDPC codeaccording to an embodiment.

A spatially-coupled LDPC code 200 has element matrixes 201 to 205arranged three by three in the column direction. The element matrixes201 are arranged at each of both ends of the spatially-coupled LDPC code200, the element matrixes 202 are arranged inside the element matrixes201 and the element matrixes 203 are arranged inside the elementmatrixes 202. The element matrixes 204 are arranged inside the elementmatrixes 203 and the element matrixes 205 are arranged inside theelement matrixes 204, that is, in the center of the spatially-coupledLDPC code 200.

The column weight of the element matrix 201 is larger than the columnweight of the element matrix 202, the column weight of the elementmatrix 202 is larger than the column weight of the element matrix 203,the column weight of the element matrix 203 is larger than the columnweight of the element matrix 204, and the column weight of elementmatrix 204 is greater than the column weight of element matrix 205. Ingeneral, in the error correction process using the spatially-coupledLDPC code, the coding rate at the end portion of a signal bit string issmaller than that in the central portion thereof. Thus, the errorcorrection first proceeds from the end portion. The spatially-coupledLDPC code 200 provides the strengthened effect of correcting an errorfrom the end portion of the signal bit string when the element matrix201 having the largest column weight is arranged at both ends and theelement matrix 205 having the smallest column weight is arranged at thecentral portion, which makes it possible to achieve the error correctionwith fewer processing times.

However, in the spatially-coupled LDPC code 200, since the column weightdifference between the element matrix 201 and the element matrix 205 islarge, a shift in processing time due to the column weight differencemay occur and there is a possibility that a circuit for executing theerror correction process becomes complicated.

(Spatially-Coupled LDPC Code According to First Embodiment)

FIG. 3 is a view illustrating an example of a spatially-coupled LDPCcode according to a first embodiment.

A spatially-coupled LDPC code 1 has element matrixes 11 and 12 arrangedthree by three in the column direction. The element matrixes 11 denotedby H_(1,1) to H_(1,3) and H_(4,1) to H_(4,3) are arranged at both endsof the spatially-coupled LDPC code 1, respectively, and the elementmatrixes 12 denoted by H_(2,1) to H_(2,3) H_(3,1) to H_(3,3) arearranged inside the element matrixes 11, that is, at the center of thespatially-coupled LDPC code 1.

The column weight of the element matrix 11 is substantially the same asthe column weight of the element matrix 12. The number of rows of theelement matrix 11 is the same as the number of rows of the elementmatrix 12. However, the number of columns of the element matrix 11 islarger than the number of columns of the element matrix 12. When thenumber of columns of the element matrix 11 located near both ends havinga large contribution to the error correction process is larger than thenumber of columns of the element matrix 12 positioned at the center, thespatially-coupled LDPC code 1 may use the effect of spatial coupling toimprove the characteristics of the error correction process.

(Spatially-Coupled LDPC Code According to Second Embodiment)

FIG. 4 is a view illustrating a spatially-coupled LDPC code according toa second embodiment.

A spatially-coupled LDPC code 2 is a generalized spatially-coupled LDPCcode 1 according to the first embodiment. The spatially-coupled LDPCcode 2 has element matrixes of L columns. That is, the spatially-coupledLDPC code 2 has W element matrixes 21 denoted by H_(1,1) to H_(1,W), Welement matrixes 22 denoted by H_(2,1) to H_(2,W), and W elementmatrixes 23 denoted by H_(3,1) to H_(3,W). The spatially-coupled LDPCcode 2 further has W element matrixes 2(L−2) denoted by H_((L−2),1) toH_((L−2),W), W element matrixes 2(L−1) denoted by H_((L−1),1) toH_((L−1),W), and W element matrixes 2L denoted by H_(L,1) to H_(L,W).The column weights of the element matrixes 21 to 2L are substantiallythe same and the number of rows of the element matrixes 21 to 2L is thesame.

The spatially-coupled LDPC code 2 is formed such that the number ofcolumns of the element matrix arranged in the column spaced apart by afirst distance from the right end becomes equal to the number of columnsof the element matrix arranged in the column spaced apart by the firstdistance from the left end. For example, the number of columns of theelement matrix 21 and the number of columns of the element matrix 2L areequal to each other, the number of columns of the element matrix 22 andthe number of columns of the element matrix 2(L−1) are equal to eachother, and the number of columns of the element matrix 23 and the numberof columns of the element matrix 2(L−2) are equal to each other. Bymaking the number of columns of element matrixes arranged in the columnspaced apart by the same distance from the left and right ends equal toeach other, the spatially-coupled LDPC code 2 may be used to execute abit string error correction process to equalize processing speeds of theLSB side and the MSB side of the bit string.

In addition, the number of columns of the element matrix 21 is largerthan the number of columns of the element matrix 22, and the number ofcolumns of the element matrix 22 is larger than the number of columns ofthe element matrix 23. Similarly, the spatially-coupled LDPC code 2 isformed such that the number of rows of the element matrix decreases fromthe end toward the center.

In addition, the number of columns of the element matrix included in thespatially-coupled LDPC code 2 is formed to be an integral multiple ofthe number of columns of the element matrix having the smallest numberof rows. For example, when the element matrix is arranged over fourcolumns, the ratio of the number of columns of the element matrix may beset to 2:1:1:2, 3:1:1:3, etc. When the number of columns of the elementmatrix is formed to be an integral multiple of the number of columns ofthe element matrix having the smallest number of columns, thespatially-coupled LDPC code 2 may be easily generated.

(Spatially-Coupled LDPC Code According to Third Embodiment)

FIG. 5 is a view illustrating a spatially-coupled LDPC code according toa third embodiment.

Similarly to the spatially-coupled LDPC code 2, a spatially-coupled LDPCcode 3 has element matrixes of L columns. However, the element matrix ofthe spatially-coupled LDPC code 3 is different from the element matrixof the spatially-coupled LDPC code 2 in that the former has the samenumber of columns and the different number of rows.

Specifically, the spatially-coupled LDPC code 3 has W element matrixes311 to 31W denoted by H_(1,1) to H_(1,W), W element matrixes 321 to 32Wdenoted by H_(2,1) to H_(2,W), and W element matrixes 331 to 33W denotedby H_(3,1) to H_(3,W). The spatially-coupled LDPC code 3 further has Welement matrixes 3(L−2)1 to 3(L−2)W denoted by H_((L−2),1) toH_((L−2),W), W element matrixes 3(L−1)1 to 3(L−1)W denoted byH_((L−1),1) to H_((L−1),W), and W element matrixes 3L1 to 3LW denoted byH_(L,1) to H_(L,W). The column weights of the element matrixes 311 to31W and so on of the spatially-coupled LDPC code 3 are substantially thesame. The number of columns of the element matrixes 311 to 31W and so onof the spatially-coupled LDPC code 3 is the same.

The spatially-coupled LDPC code 3 is formed such that the number of rowsof the element matrix arranged in the column spaced apart by a seconddistance from the upper end becomes equal to the number of rows of theelement matrix arranged in the column spaced apart by the seconddistance from the lower end. For example, the number of rows of theelement matrix 312 is larger than the number of rows of the elementmatrix 311, the number of rows of the element matrix 322 is larger thanthe number of rows of the element matrix 321, and the number of rows ofthe element matrix 332 is larger than the number of rows of the elementmatrix 331. In the meantime, the number of rows of the element matrix3(L−2)2 is smaller than the number of rows of the element matrix3(L−2)1, the number of rows of the element matrix 3(L−1)2 is smallerthan the number of rows of the element matrix 3(L−1)1, and the number ofrows of the element matrix 3L2 is smaller than the number of rows of theelement matrix 3L1. By making the number of columns of element matrixesarranged in the row spaced apart by the same distance from the upper andlower ends equal to each other, the spatially-coupled LDPC code 3 may beused to execute a bit string error correction process to equalizeprocessing speeds of the LSB side and the MSB side of the bit string.

In the spatially-coupled LDPC code 3, the number of rows of the elementmatrixes arranged in the same row is the same. For example, the elementmatrix 312 and the element matrix 321 have the same number of rows, andthe element matrix 322 and the element matrix 331 have the same numberof rows. In addition, the element matrix 3(L−2)2 and the element matrix3(L−1)1 have the same number of rows, and the element matrix 3(L−1)2 andthe element matrix 3L1 have the same number of rows.

In addition, since the spatially-coupled LDPC code 3 is formed such thatthe number of rows of the element matrix increases from the end towardthe center, the constraint condition of the corresponding signal bitstring more strongly acts. That is, the spatially-coupled LDPC code 3may improve the characteristics of the correction process by setting thenumber of rows of the element matrix located at the center to be largerthan the number of rows of the element matrix located at the end.

In addition, the number of rows of the element matrix included in thespatially-coupled LDPC code 3 is formed to be an integral multiple ofthe number of rows of the element matrix having the smallest number ofrows. For example, when the element matrix is arranged over four rows,the ratio of the number of rows of the element matrix may be set to2:1:1:2, 3:1:1:3, etc. When the number of rows of the element matrix isformed to be an integral multiple of the number of rows of the elementmatrix having the smallest number of rows, the spatially-coupled LDPCcode 3 may be easily generated.

(Spatially-Coupled LDPC Code According to Fourth Embodiment)

FIG. 6 is a view illustrating a spatially-coupled LDPC code according toa fourth embodiment.

Similarly to the spatially-coupled LDPC code 2, a spatially-coupled LDPCcode 4 has element matrixes of L columns. However, the element matrix ofthe spatially-coupled LDPC code 4 is different from the element matrixof the spatially-coupled LDPC code 2 in that the former has thedifferent number of columns and the different number of rows.

Specifically, the spatially-coupled LDPC code 4 has W element matrixes411 to 41W denoted by H_(1,1) to H_(1,W), W element matrixes 421 to 42Wdenoted by H_(2,1) to H₂″ and W element matrixes 431 to 43W denoted byH_(3,1) to H_(3,W). The spatially-coupled LDPC code 4 further has Welement matrixes 4(L−2)1 to 4(L−2)W denoted by H_((L−2),1) toH_((L−2),W), W element matrixes 4(L−1)1 to 4(L−1)W denoted byH_((L−1),1) to H_((L−1),W), and W element matrixes 4L1 to 4LW denoted byH_(L,1) to H_(L,W). The column weights of the element matrixes 411 to41W and so on of the spatially-coupled LDPC code 4 are substantially thesame.

The number of rows of the element matrix 412 is larger than the numberof rows of the element matrix 411, the number of rows of the elementmatrix 422 is larger than the number of rows of the element matrix 421,and the number of rows of the element matrix 432 is larger than thenumber of rows of the element matrix 431. In the meantime, the number ofrows of the element matrix 4(L−2)2 is smaller than the number of rows ofthe element matrix 4(L−2)1, the number of rows of the element matrix4(L−1)2 2 is smaller than the number of rows of the element matrix4(L−1)1, and the number of rows of the element matrix 4L2 is smallerthan the number of rows of the element matrix 4L1. Similarly, thespatially-coupled LDPC code 4 is formed such that the number of rows ofthe element matrix increases from the end towards the center.

In the spatially-coupled LDPC code 4, the number of rows of the elementmatrixes arranged in the same row is the same. For example, the elementmatrix 412 and the element matrix 421 have the same number of rows, andthe element matrix 422 and the element matrix 431 have the same numberof rows. In addition, the element matrix 4(L−2)2 and the element matrix4(L−1)1 have the same number of rows, and the element matrix 4(L−1)2 andthe element matrix 4L1 have the same number of rows.

The spatially-coupled LDPC code 4 may improve the characteristics of thecorrection process by being formed such that the number of columns ofthe element matrix decreases from the end toward the center and thenumber of rows of the element matrix increases from the end toward thecenter.

(Method of Generating Spatially-Coupled LDPC Code According toEmbodiment)

FIG. 7 is a view for explaining a method of generating thespatially-coupled LDPC code 4. In FIG. 7, K is the number of columns ofthe reference element matrix, L is the length of spatial coupling, N isa difference between the number of columns and the number of rows of thereference element matrix, W is the width of spatial coupling, s₁ tos_(L,W−1) are shape variables in the column direction, and t₁ to t_(L)are shape variables in the column direction. Each of the shape variabless_(l) and t_(i) shows the following relationship.

$\begin{matrix}{\lbrack {{Eq}.\mspace{14mu} 1} \rbrack\mspace{675mu}} & \; \\{{{\sum\limits_{l = 1}^{L}s_{l}} = L},{{\sum\limits_{j = 1}^{L + W - 1}t_{j}} = {L + W - 1.}}} & (1)\end{matrix}$

Since each of the shape variables s_(l) and t_(i) shows the relationshipof Equation (1), the coding rate r_(SC) of the spatially-coupled LDPCcode 4 is uniquely determined by the constants K, L, N and W regardlessof values of the shape variables s_(l) and t_(i), as expressed by thefollowing equation (2).

$\begin{matrix}{\lbrack {{Eq}.\mspace{14mu} 2} \rbrack\mspace{675mu}} & \; \\{r_{SC} = {\frac{{LK} - {( {W - 1} )( {N - K} )}}{LN}.}} & (2)\end{matrix}$

The bit erasure probability p^((I)) in the entire codeword when aniterative decoding process is executed (I) times is expressed by thefollowing equation (3).

$\begin{matrix}{\lbrack {{Eq}.\mspace{14mu} 3} \rbrack\mspace{675mu}} & \; \\{p^{(I)} = {\frac{ɛ}{L}{\sum\limits_{l = 1}^{L}{s_{l}{\sum\limits_{w = 1}^{W}{{v_{lw}( q_{lw}^{({l - 1})} )}.}}}}}} & (3)\end{matrix}$

Where, ε is the initial value of the bit erasure probability p^((I)),and q^((I)) _(lw) is expressed by the following equation (4). q^((I))_(lw) is calculated alternately with p^((I−1)) _(lw) expressed by thefollowing equation (4) while sequentially incrementing the decodingcount I from 0.

$\begin{matrix}{\lbrack {{Eq}.\mspace{14mu} 4} \rbrack\mspace{675mu}} & \; \\{{p_{lw}^{(I)} = {ɛ\;{\lambda_{lw}( q_{lw}^{({I - 1})} )}{\prod\limits_{w^{\prime} \neq w}\;{v_{{lw}^{\prime}}( q_{{lw}^{\prime}}^{({I - 1})} )}}}},{q_{lw}^{(I)} = {1 - {{\rho_{lw}( {1 - p_{lw}^{(I)}} )}{\prod\limits_{l^{\prime} \neq l}\;{{h_{l^{\prime}w}( {1 - p_{l^{\prime}w}^{(I)}} )}.}}}}}} & (4)\end{matrix}$

In Equations (3) and (4), v_(lw)(x) is a function indicating adistribution of the ratio of “1” in each row of the element matrix, thatis, a node distribution of column weights, and is expressed by thefollowing equation (5).

$\begin{matrix}{\lbrack {{Eq}.\mspace{14mu} 5} \rbrack\mspace{675mu}} & \; \\{{{v_{lw}(x)} = {{v_{{lw},1}x} + {v_{{lw},2}x^{2}} + \ldots + {v_{{lw},i}x^{i}} + \ldots}}\mspace{14mu},{{\sum\limits_{i}v_{{lw},i}} = 1}} & (5)\end{matrix}$

In Equation (4), h_(lw)(x) is a function indicating a distribution ofthe ratio of “1” in each column of the element matrix, that is, a nodedistribution of column weights, and is expressed by the followingequation (6).

$\begin{matrix}{\lbrack {{Eq}.\mspace{14mu} 6} \rbrack\mspace{675mu}} & \; \\{{h_{lw}(x)} = {{{h_{{lw},1}x} + {h_{{lw},2}x^{2}} + \ldots + {h_{{lw},i}x^{i}} + {\ldots\mspace{14mu}.{\sum\limits_{i}h_{{lw},i}}}} = 1}} & (6)\end{matrix}$

In Equation (4), λ_(lw)(x) is a function indicating a distribution whenthe ratio of “1” in each row of the element matrix is converted into theratio of edge in a protograph, that is, an edge distribution of columnweights, and is expressed by the following equation (7).

$\begin{matrix}{\lbrack {{Eq}.\mspace{14mu} 7} \rbrack\mspace{675mu}} & \; \\{{{\lambda_{lw}(x)} = {{\lambda_{{lw},1}x} + {\lambda_{{lw},2}x} + \ldots + {\lambda_{{lw},i}x^{i - 1}} + \ldots}}\mspace{14mu},{{\sum\limits_{i}\lambda_{{lw},i}} = 1}} & (7)\end{matrix}$

FIG. 8 is a view illustrating an example of a protograph which is alsoreferred to as a Tanner graph. In FIG. 8, a circle corresponds to therow of a parity check matrix, a square corresponds to the column of theparity check matrix, and a straight line connecting the circle and thesquare is called edge and corresponds to “1” included in the paritycheck matrix.

In Equation (4), ρ_(lw)(x) is a function indicating a distribution whenthe ratio of “1” in each column of the element matrix is converted intothe ratio of edge in a protograph, that is, an edge distribution ofcolumn weights, and is expressed by the following equation (8).

$\begin{matrix}{\lbrack {{Eq}.\mspace{14mu} 8} \rbrack\mspace{675mu}} & \; \\{{\rho_{lw}(x)} = {{{\rho_{{lw},1}x} + {\rho_{{lw},2}x} + \ldots + {\rho_{{lw},i}x^{i - 1}} + {\ldots\mspace{14mu}.{\sum\limits_{i}\rho_{{lw},i}}}} = 1}} & (8)\end{matrix}$

The function v_(lw)(x) expressed by Equation (5) and the function λ_(lw)(x) expressed by Equation (7), and the function h_(lw)(x) expressed byEquation (6) and the function ρ_(lw)(x) expressed by Equation (8), arerespectively correlated as expressed by the following equation (9). Thatis, the function v_(lw)(x) and the function λ_(lw) (x) correlate witheach other and the function h_(lw)(x) and the function ρ_(lw)(x)correlate with each other.

$\begin{matrix}{\lbrack {{Eq}.\mspace{14mu} 9} \rbrack\mspace{675mu}} & \; \\{{v_{{lw},i} = \frac{\lambda_{{lw},i}/i}{\sum\limits_{j}{\lambda_{{lw},j}/j}}},{h_{{lw},i} = {\frac{\rho_{{lw},i}/i}{\sum\limits_{j}{\rho_{{lw},j}/j}}.}}} & (9)\end{matrix}$

The four functions of Equations (5) to (8) correlate with each other bybeing given a constraint condition expressed by the following equation(10) and only the function v_(lw)(x) expressed by Equation (5) is anindependent function. Equation (1) includes the first equation describedabove and the second equation described below.

$\begin{matrix}{\lbrack {{Eq}.\mspace{14mu} 10} \rbrack\mspace{644mu}} & \; \\{{\overset{\_}{h_{lw}} = {\frac{s_{l}N}{t_{l + w - 1}( {N - K} )}{\sum\limits_{i}{v_{{lw},i} \cdot i}}}},{{h_{lw}(x)} = {{( {{\overset{\_}{h_{lw}}} - \overset{\_}{h_{lw}}} )x^{\lfloor\overset{\_}{h_{lw}}\rfloor}} + {( {\overset{\_}{h_{lw}} - \lfloor \overset{\_}{h_{lw}} \rfloor} ){x^{\lfloor\overset{\_}{h_{lw}}\rfloor}.}}}}} & (10)\end{matrix}$

The shape variables s_(l) and t_(i) are determined by an optimizationalgorithm such as dichotomy, a genetic algorithm or the like such thatthe erasure probability p^((I)) is equal to or less than a desiredthreshold value of about 10⁻¹⁵ in the optical fiber communication untilthe decoding number I reaches the maximum value limit Imax of about 10to 1000. For example, the values of the shape variables s_(l) and t_(i)that fix the coding rate r_(SC), the length L of the spatial couplingand the width W of the spatial coupling to a desired value and maximizethe initial value E of the erasure probability may be searched, togetherwith the value of the function v_(lw)(x), using the optimizationalgorithm.

The spatially-coupled LDPC code 2 may be calculated in the same manneras the spatially-coupled LDPC code 4 except that t_(i) is set to “1” inEquations (1) to (10). In addition, the spatially-coupled LDPC code 3may be calculated in the same manner as the spatially-coupled LDPC code4 except that s_(l) is set to “1” in Equations (1) to (10).

(Apparatus Using Spatially-Coupled LDPC According to Embodiment)

FIG. 9A is a view illustrating a communication system, FIG. 9B is aninternal circuit diagram of a transmitter illustrated in FIG. 9A, FIG.9C is an internal circuit diagram of a receiver illustrated in FIG. 9A,and FIG. 9D is a view illustrating a storage system

The communication system 6 includes a transmitter 61, a transmissionline 62, and a receiver 63. The transmitter 61 and the receiver 63 arean example of a transmission apparatus according to an embodiment. Thetransmitter 61 encodes a bit string u1 corresponding to an input signalusing a generation matrix G to generate a coded bit string c1 andtransmits a transmission signal TS, which is generated by performing apredetermined characteristic compensation process for the coded bitstring c1, to the receiver 63 via the transmission line 62. Afterperforming a waveform shaping process and a carrier phase restoringprocess for the reception signal RS, the receiver 63 uses a parity checkmatrix H, which is the spatially-coupled LDPC according to anembodiment, to perform an error correction process for a bit string c2corresponding to the reception signal RS to restore a bit string u2. Thegeneration matrix G and the parity check matrix H have a relationship of“GHT=0.”

The transmitter 61 includes an encoding circuit 611 and apre-equalization circuit 612. The encoding circuit 611 uses thegenerator matrix G to generate the encoded bit string c1 from the bitstring u1 corresponding to the input signal and outputs the generatedencoded bit string c1 to the pre-equalization circuit 612. Thepre-equalization circuit 612 performs various characteristiccompensation processes such as wavelength dispersion compensation andfrequency offset compensation for the coded bit string c1 to generatethe transmission signal TS, and transmits the generated transmissionsignal TS to the receiver 63 via the transmission path 62.

The receiver 63 includes an equalization circuit 631, a carrier phaserestoration circuit 632, and a decoding circuit 633. The equalizationcircuit 631 and the carrier phase restoration circuit 632 form areceiving circuit that receives a reception signal RD indicating anencoded bit string. The equalization circuit 631 performs variouswaveform shaping processes such as wavelength dispersion compensation,frequency offset compensation, polarization mode dispersioncompensation, and waveform distortion compensation for the receptionsignal RD. The carrier phase restoration circuit 632 detects a phasedifference between the reception signal and a clock, and restores thereception signal indicating the coded bit string c2 based on thedetected phase difference. The decoding circuit 633 uses the paritycheck matrix H to perform an error correction process for the bit stringc2 corresponding to the restored reception signal to generate acorrected bit string u2. The decoding circuit 633 outputs the correctedbit string. The bit string u2 generated by the decoding circuit 633corresponds to the bit string u1 input to the transmitter 61.

The storage system 7 includes an encoding circuit 71, a storage medium72, and a decoding circuit 73. The encoding circuit 71 and the decodingcircuit 73 are another example of the transmission apparatus accordingto the embodiment. The encoding circuit 71 uses the generation matrix Gto encode a write bit string to generate the encoded bit string c1 andwrites the generated encoded bit string c1 in the storage medium 72. Thestorage medium 72 is, for example, a semiconductor memory and stores theencoded bit string c1 written by the encoding circuit 71. The decodingcircuit 73 reads the encoded bit string c2 stored in the storage medium72 from the storage medium 72 and uses the parity check matrix H, whichis the spatially-coupled LDPC according to the embodiment, to execute anerror correction process to restore a read bit string r.

(Modification of Spatially-Coupled LDPC Code According to Embodiment)

The spatially-coupled LDPC code according to the embodiment is notlimited to the disclosed spatially-coupled LDPC code but may includeother spatially-coupled codes such as a spatially-couplerepeat-accumulate (SC-RA) code described in Non-Patent Document 1. Thespatially-coupled RA code has the characteristic in the error correctioncapability of the spatially-coupled LDPC code and is easy to process theencoding, and is suitable for practical use in applying the effect ofthe spatial coupling.

FIG. 10 is a view illustrating a spatially-coupled RA code according toan embodiment.

A spatially-coupled RA code 5 has a spatial coupling unit 51 and astorage unit 52 which is also called an accumulation unit. Since thespatial coupling unit 51 has the same configuration as thespatially-coupled LDPC code 4, detailed explanation thereof will not berepeated.

The storage unit 52 includes “1” arranged on a diagonal line and “1”sarranged below and adjacent to the “1” arranged stepwise on the diagonalline. In addition, the storage unit 52 further includes “1” stepwisearranged spaced apart from “1” arranged on the diagonal line. Although“1” arranged below and adjacent to the “1” arranged stepwise on thediagonal line are included in the storage unit 52, “1” may be arrangedabove and adjacent to the “1” arranged stepwise on the diagonal line. Inaddition, although the storage unit 52 further includes “1” stepwisearranged spaced apart from the “1” arranged on the diagonal line, the“1” stepwise arranged spaced apart from the “1” arranged on the diagonalline may be omitted.

(Method of Generating Spatially-Coupled LDPC Code According toModification)

FIG. 11 is a view for explaining a method of generating aspatially-coupled RA code 5. The constants K, L, N, W and variables s₁to s_(L+W−1) and t₁ to t_(L) illustrated in FIG. 11 are the same as theconstants K, L, N, W and variables s₁ to s_(L+W−1) and t₁ to t_(L)illustrated in FIG. 7, respectively, and therefore, detailed explanationthereof will not be repeated.

Hereinafter, differences from the method of generating thespatially-coupled LDPC code 4 described with reference to FIG. 7 will bedescribed. The coding rate r_(SC) is defined by the following equation(11) instead of Equation (2).

$\begin{matrix}{\lbrack {{Eq}.\mspace{14mu} 11} \rbrack\mspace{644mu}} & \; \\{r_{SC} = {\frac{LK}{{LK} + {( {L + W - 1} )( {N - K} )}}.}} & (11)\end{matrix}$

In addition, the bit erasure probability p^((I)) in the entire codewordwhen the iterative decoding process is performed (I) times is defined bythe following equations (12) and (13) instead of Equations (3) and (4).

$\begin{matrix}{\lbrack {{Eq}.\mspace{14mu} 12} \rbrack\mspace{644mu}} & \; \\{p^{(I)} = {{r_{SC}\frac{ɛ}{L}{\sum\limits_{l = 1}^{L}{s_{l}{\prod\limits_{w = 1}^{W}\;{v_{lw}( q_{lw}^{({I - 1})} )}}}}} + {( {1 - r_{SC}} )\frac{ɛ}{L + W - 1}{\sum\limits_{j = 1}^{L + W - 1}{t_{j}{Q_{j}^{{({I - 1})}^{d_{p}}}.}}}}}} & (12) \\{\lbrack {{Eq}.\mspace{14mu} 13} \rbrack\mspace{644mu}} & \; \\{{p_{lw}^{(I)} = {ɛ\;{\lambda_{lw}( q_{lw}^{({I - 1})} )}{\prod\limits_{w^{\prime} \neq w}\;{v_{{lw}^{\prime}}( q_{{lw}^{\prime}}^{({I - 1})} )}}}},{q_{lw}^{(I)} = {1 - {( {1 - p_{l + w - 1}^{(I)}} )^{d_{p} - 1}{\rho_{lw}( {1 - p_{lw}^{(I)}} )}{\prod\limits_{l^{\prime} \neq l}\;{h_{l^{\prime}w}( {1 - p_{l^{\prime}w}^{(I)}} )}}}}},{P_{l + w - 1}^{(I)} = {ɛ( {1 - Q_{l + w - 1}^{({I - 1})}} )}^{d_{p} - 1}},{Q_{l + w - 1}^{(I)} = {1 - {( {1 - P_{l + w - 1}^{(I)}} )^{d_{p} - 1}{\prod\limits_{l^{\prime}}\;{h_{l^{\prime}w}{( {1 - p_{l^{\prime}w}^{(I)}} ).}}}}}}} & (13)\end{matrix}$

In Equation (13), P^((I)) _(1+W−1) and Q^((I)) _(1+W−1) are equationscorresponding to the storage unit 52, as expressed by P_(j) and Q_(j) ina protograph illustrated in FIG. 12.

In addition, in order to correlate the four functions of Equations (5)to (8), the following equation (14) is used instead of the firstequation of Equation (10).

$\begin{matrix}{\lbrack {{Eq}.\mspace{14mu} 14} \rbrack\mspace{644mu}} & \; \\{{\overset{\_}{h_{lw}} = {\frac{s_{l}K}{t_{l + w - 1}( {N - K} )}{\sum\limits_{i}{v_{{lw},i} \cdot i}}}},} & (14)\end{matrix}$

The shape variables s_(l) and t_(i) of the spatially-coupled RA code 5are searched, together with the value of the function v_(lw)(x), by theoptimization algorithm using Equations (1), (5) to (9), the secondequation of Equation (10), and (11) to (14).

An apparatus using the spatially-coupled LDPC according to themodification has the same configuration as the apparatus described withreference to FIG. 9 except that the former encodes a bit string usingthe spatially-coupled LDPC according to the modification without usingthe generation matrix G, and therefore, detailed explanation thereofwill not be repeated.

All examples and conditional language recited herein are intended forpedagogical purposes to aid the reader in understanding the inventionand the concepts contributed by the inventor to furthering the art, andare to be construed as being without limitation to such specificallyrecited examples and conditions, nor does the organization of suchexamples in the specification relate to an illustrating of thesuperiority and inferiority of the invention. Although the embodimentsof the present invention have been described in detail, it should beunderstood that the various changes, substitutions, and alterationscould be made hereto without departing from the spirit and scope of theinvention.

What is claimed is:
 1. A transmission apparatus comprising: a receivingcircuit configured to receive a reception signal indicating a coded bitstring; a decoding circuit configured to decode and correct the bitstring when an error is detected in reception signal by using aspatially-coupled low density parity check code constituted by arrangingelement matrixes stepwise in a diagonal direction, a parity check matrixof the spatially-coupled low density parity check code including atleast one element matrix having at least one of a number of rows and anumber of columns different from a number of rows and a number ofcolumns of other element matrixes when each sparse matrix constitutingthe parity check matrix is regarded as an element matrix, the coded bitstring is decoded and corrected when the reception signal is received toprovide a corrected bit string, and the corrected bit string is output,wherein a number of columns of an element matrix arranged in a column ata center of the spatially-coupled low density parity check code issmaller than a number of columns of an element matrix arranged in acolumn at an end of the spatially-coupled low density parity check code.2. The transmission apparatus according to claim 1, wherein the numberof columns of the element matrix included in the spatially-coupled lowdensity parity check code is an integral multiple of a number of columnsof an element matrix having a smallest number of rows.
 3. Thetransmission apparatus according to claim 1, wherein a number of columnsof an element matrix arranged in a column spaced apart by a firstdistance from a right end of the spatially-coupled low density paritycheck code is equal to a number of columns of an element matrix arrangedin the column spaced apart by the first distance from a left end of thespatially-coupled low density parity check code.
 4. The transmissionapparatus according to claim 1, wherein a number of rows of the elementmatrix arranged in a row at the center of the spatially-coupled lowdensity parity check code is larger than a number of rows of the elementmatrix arranged in a row at the end of the spatially-coupled low densityparity check code.
 5. The transmission apparatus according to claim 4,wherein a number of rows of the element matrix included in thespatially-coupled low density parity check code is an integral multipleof a number of rows of an element matrix having a smallest number ofrows.
 6. The transmission apparatus according to claim 4, wherein anumber of rows of an element matrix arranged in a row spaced apart by asecond distance from an upper end of the spatially-coupled low densityparity check code is equal to a number of columns of an element matrixarranged in a row spaced apart by the second distance from a lower endof the spatially-coupled low density parity check code.
 7. Thetransmission apparatus according to claim 1, wherein thespatially-coupled low density parity check code further includes astorage unit including “1” arranged on a diagonal line and “1” arrangedadjacent one of above and below the “1” arranged on the diagonal line.8. An error correction method, comprising: receiving a reception signalindicating a coded bit string; generating a parity check matrix of aspatially-coupled low density parity check code including at least oneelement matrix having at least one of a number of rows and a number ofcolumns different from a number of rows and a number of columns of otherelement matrixes when each sparse matrix constituting the parity checkmatrix is regarded as an element matrix; decoding and correcting the bitstring when an error is detected in reception signal by using thespatially-coupled low density parity check code constituted by arrangingelement matrixes stepwise in a diagonal direction, a parity check matrixof the spatially-coupled low density parity check code including atleast one element matrix having at least one of a number of rows and anumber of columns different from a number of rows and a number ofcolumns of other element matrixes when each sparse matrix constitutingthe parity check matrix is regarded as an element matrix, the coded bitstring is decoded and corrected when the reception signal is received toprovide a corrected bit string; and outputting the corrected bit stringis output, wherein a number of columns of an element matrix arranged ina column at a center of the spatially-coupled low density parity checkcode is smaller than a number of columns of an element matrix arrangedin a column at an end of the spatially-coupled low density parity checkcode.
 9. The error correction method according to claim 8, wherein thenumber of columns of the element matrix included in thespatially-coupled low density parity check code is an integral multipleof a number of columns of an element matrix having a smallest number ofrows.